Problem:
A positive integer divisor of is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as , where and are relatively prime positive integers. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The prime factorization of is
This yields a total of
divisors of
In order to produce a perfect square divisor, there must be an even exponent for each number in the prime factorization. Note that the divisor can't have any factors of and in the prime factorization because there is only one of each in . Thus, there are
(For , you can have , or , etc.)
The probability that the divisor chosen is a perfect square is
The problems on this page are the property of the MAA's American Mathematics Competitions